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Properties

Label 900.2.s.d.349.7
Level $900$
Weight $2$
Character 900.349
Analytic conductor $7.187$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,2,Mod(49,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.1333317747165888577536.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 349.7
Root \(1.15347 + 0.818235i\) of defining polynomial
Character \(\chi\) \(=\) 900.349
Dual form 900.2.s.d.49.7

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.28535 + 1.16098i) q^{3} +(4.32643 + 2.49787i) q^{7} +(0.304233 + 2.98453i) q^{9} +(-1.99787 + 3.46041i) q^{11} +(-1.33642 + 0.771582i) q^{13} -6.99574i q^{17} +2.25667 q^{19} +(2.66098 + 8.23355i) q^{21} +(-6.75116 + 3.89778i) q^{23} +(-3.07395 + 4.18937i) q^{27} +(3.08304 - 5.33998i) q^{29} +(0.271582 + 0.470394i) q^{31} +(-6.58543 + 2.12833i) q^{33} -6.25240i q^{37} +(-2.61356 - 0.559811i) q^{39} +(-0.0979532 - 0.169660i) q^{41} +(-0.0747616 - 0.0431636i) q^{43} +(3.31658 + 1.91483i) q^{47} +(8.97869 + 15.5515i) q^{49} +(8.12194 - 8.99195i) q^{51} -4.19164i q^{53} +(2.90060 + 2.61995i) q^{57} +(3.51278 + 6.08432i) q^{59} +(1.45470 - 2.51962i) q^{61} +(-6.13873 + 13.6723i) q^{63} +(7.76470 - 4.48295i) q^{67} +(-13.2028 - 2.82798i) q^{69} +8.79130 q^{71} +2.28650i q^{73} +(-17.2873 + 9.98082i) q^{77} +(-6.32211 + 10.9502i) q^{79} +(-8.81488 + 1.81599i) q^{81} +(-12.0911 - 6.98082i) q^{83} +(10.1624 - 3.28437i) q^{87} +10.3577 q^{89} -7.70924 q^{91} +(-0.197042 + 0.919921i) q^{93} +(-8.08758 - 4.66936i) q^{97} +(-10.9355 - 4.90993i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{9} + 6 q^{11} + 16 q^{19} + 26 q^{21} + 18 q^{29} - 4 q^{31} + 34 q^{39} + 18 q^{41} + 18 q^{49} + 6 q^{51} + 30 q^{59} + 2 q^{61} - 18 q^{69} - 48 q^{71} - 14 q^{79} - 62 q^{81} - 12 q^{89}+ \cdots - 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.28535 + 1.16098i 0.742095 + 0.670294i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.32643 + 2.49787i 1.63524 + 0.944105i 0.982441 + 0.186573i \(0.0597379\pi\)
0.652797 + 0.757533i \(0.273595\pi\)
\(8\) 0 0
\(9\) 0.304233 + 2.98453i 0.101411 + 0.994845i
\(10\) 0 0
\(11\) −1.99787 + 3.46041i −0.602380 + 1.04335i 0.390080 + 0.920781i \(0.372447\pi\)
−0.992460 + 0.122571i \(0.960886\pi\)
\(12\) 0 0
\(13\) −1.33642 + 0.771582i −0.370656 + 0.213998i −0.673745 − 0.738964i \(-0.735315\pi\)
0.303089 + 0.952962i \(0.401982\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.99574i − 1.69672i −0.529424 − 0.848358i \(-0.677592\pi\)
0.529424 − 0.848358i \(-0.322408\pi\)
\(18\) 0 0
\(19\) 2.25667 0.517715 0.258857 − 0.965916i \(-0.416654\pi\)
0.258857 + 0.965916i \(0.416654\pi\)
\(20\) 0 0
\(21\) 2.66098 + 8.23355i 0.580674 + 1.79671i
\(22\) 0 0
\(23\) −6.75116 + 3.89778i −1.40771 + 0.812744i −0.995167 − 0.0981929i \(-0.968694\pi\)
−0.412546 + 0.910937i \(0.635360\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.07395 + 4.18937i −0.591582 + 0.806245i
\(28\) 0 0
\(29\) 3.08304 − 5.33998i 0.572506 − 0.991609i −0.423802 − 0.905755i \(-0.639305\pi\)
0.996308 − 0.0858540i \(-0.0273619\pi\)
\(30\) 0 0
\(31\) 0.271582 + 0.470394i 0.0487775 + 0.0844852i 0.889383 − 0.457162i \(-0.151134\pi\)
−0.840606 + 0.541647i \(0.817801\pi\)
\(32\) 0 0
\(33\) −6.58543 + 2.12833i −1.14638 + 0.370495i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.25240i − 1.02789i −0.857824 − 0.513944i \(-0.828184\pi\)
0.857824 − 0.513944i \(-0.171816\pi\)
\(38\) 0 0
\(39\) −2.61356 − 0.559811i −0.418504 − 0.0896414i
\(40\) 0 0
\(41\) −0.0979532 − 0.169660i −0.0152977 − 0.0264964i 0.858275 − 0.513190i \(-0.171536\pi\)
−0.873573 + 0.486693i \(0.838203\pi\)
\(42\) 0 0
\(43\) −0.0747616 − 0.0431636i −0.0114010 − 0.00658239i 0.494289 − 0.869298i \(-0.335429\pi\)
−0.505690 + 0.862715i \(0.668762\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.31658 + 1.91483i 0.483774 + 0.279307i 0.721988 − 0.691906i \(-0.243229\pi\)
−0.238214 + 0.971213i \(0.576562\pi\)
\(48\) 0 0
\(49\) 8.97869 + 15.5515i 1.28267 + 2.22165i
\(50\) 0 0
\(51\) 8.12194 − 8.99195i 1.13730 − 1.25912i
\(52\) 0 0
\(53\) − 4.19164i − 0.575766i −0.957666 − 0.287883i \(-0.907049\pi\)
0.957666 − 0.287883i \(-0.0929515\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.90060 + 2.61995i 0.384194 + 0.347021i
\(58\) 0 0
\(59\) 3.51278 + 6.08432i 0.457326 + 0.792111i 0.998819 − 0.0485942i \(-0.0154741\pi\)
−0.541493 + 0.840705i \(0.682141\pi\)
\(60\) 0 0
\(61\) 1.45470 − 2.51962i 0.186256 − 0.322604i −0.757743 − 0.652553i \(-0.773698\pi\)
0.943999 + 0.329948i \(0.107031\pi\)
\(62\) 0 0
\(63\) −6.13873 + 13.6723i −0.773407 + 1.72255i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.76470 − 4.48295i 0.948609 − 0.547680i 0.0559605 − 0.998433i \(-0.482178\pi\)
0.892649 + 0.450753i \(0.148845\pi\)
\(68\) 0 0
\(69\) −13.2028 − 2.82798i −1.58944 − 0.340449i
\(70\) 0 0
\(71\) 8.79130 1.04334 0.521668 − 0.853149i \(-0.325310\pi\)
0.521668 + 0.853149i \(0.325310\pi\)
\(72\) 0 0
\(73\) 2.28650i 0.267614i 0.991007 + 0.133807i \(0.0427203\pi\)
−0.991007 + 0.133807i \(0.957280\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −17.2873 + 9.98082i −1.97007 + 1.13742i
\(78\) 0 0
\(79\) −6.32211 + 10.9502i −0.711293 + 1.23199i 0.253080 + 0.967445i \(0.418557\pi\)
−0.964372 + 0.264549i \(0.914777\pi\)
\(80\) 0 0
\(81\) −8.81488 + 1.81599i −0.979432 + 0.201776i
\(82\) 0 0
\(83\) −12.0911 − 6.98082i −1.32717 − 0.766244i −0.342313 − 0.939586i \(-0.611210\pi\)
−0.984862 + 0.173342i \(0.944544\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.1624 − 3.28437i 1.08952 − 0.352121i
\(88\) 0 0
\(89\) 10.3577 1.09792 0.548958 − 0.835850i \(-0.315025\pi\)
0.548958 + 0.835850i \(0.315025\pi\)
\(90\) 0 0
\(91\) −7.70924 −0.808148
\(92\) 0 0
\(93\) −0.197042 + 0.919921i −0.0204324 + 0.0953914i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.08758 − 4.66936i −0.821169 − 0.474102i 0.0296505 − 0.999560i \(-0.490561\pi\)
−0.850819 + 0.525458i \(0.823894\pi\)
\(98\) 0 0
\(99\) −10.9355 − 4.90993i −1.09906 − 0.493467i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 900.2.s.d.349.7 16
3.2 odd 2 2700.2.s.d.2449.8 16
5.2 odd 4 900.2.i.e.601.3 yes 8
5.3 odd 4 900.2.i.d.601.2 yes 8
5.4 even 2 inner 900.2.s.d.349.2 16
9.2 odd 6 8100.2.d.s.649.1 8
9.4 even 3 inner 900.2.s.d.49.2 16
9.5 odd 6 2700.2.s.d.1549.1 16
9.7 even 3 8100.2.d.q.649.1 8
15.2 even 4 2700.2.i.d.1801.1 8
15.8 even 4 2700.2.i.e.1801.4 8
15.14 odd 2 2700.2.s.d.2449.1 16
45.2 even 12 8100.2.a.ba.1.4 4
45.4 even 6 inner 900.2.s.d.49.7 16
45.7 odd 12 8100.2.a.z.1.4 4
45.13 odd 12 900.2.i.d.301.2 ✓ 8
45.14 odd 6 2700.2.s.d.1549.8 16
45.22 odd 12 900.2.i.e.301.3 yes 8
45.23 even 12 2700.2.i.e.901.4 8
45.29 odd 6 8100.2.d.s.649.8 8
45.32 even 12 2700.2.i.d.901.1 8
45.34 even 6 8100.2.d.q.649.8 8
45.38 even 12 8100.2.a.y.1.1 4
45.43 odd 12 8100.2.a.x.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.2 ✓ 8 45.13 odd 12
900.2.i.d.601.2 yes 8 5.3 odd 4
900.2.i.e.301.3 yes 8 45.22 odd 12
900.2.i.e.601.3 yes 8 5.2 odd 4
900.2.s.d.49.2 16 9.4 even 3 inner
900.2.s.d.49.7 16 45.4 even 6 inner
900.2.s.d.349.2 16 5.4 even 2 inner
900.2.s.d.349.7 16 1.1 even 1 trivial
2700.2.i.d.901.1 8 45.32 even 12
2700.2.i.d.1801.1 8 15.2 even 4
2700.2.i.e.901.4 8 45.23 even 12
2700.2.i.e.1801.4 8 15.8 even 4
2700.2.s.d.1549.1 16 9.5 odd 6
2700.2.s.d.1549.8 16 45.14 odd 6
2700.2.s.d.2449.1 16 15.14 odd 2
2700.2.s.d.2449.8 16 3.2 odd 2
8100.2.a.x.1.1 4 45.43 odd 12
8100.2.a.y.1.1 4 45.38 even 12
8100.2.a.z.1.4 4 45.7 odd 12
8100.2.a.ba.1.4 4 45.2 even 12
8100.2.d.q.649.1 8 9.7 even 3
8100.2.d.q.649.8 8 45.34 even 6
8100.2.d.s.649.1 8 9.2 odd 6
8100.2.d.s.649.8 8 45.29 odd 6